Question

If $(x+\frac{1}{x})=3$, find the values of :

1) $x-\frac{1}{x}$

2) $x^2-\frac{1}{x^2}$

3) $x^4-\frac{1}{x^4}$

Answer 1

$x^2+\frac{1}{x^2}=(x+\frac{1}{x}{)}^2-2x\ast \frac{1}{x}=9-2=7$

1) $x-\frac{1}{x}=\sqrt{(x+\frac{1}{x}{)}^2-4x\ast \frac{1}{x}}=\sqrt{9-4}=\sqrt{5}$

2) $x^2-\frac{1}{x^2}=(x+\frac{1}{x})(x-\frac{1}{x})=3\ast \sqrt{5}=3\sqrt{5}$

3) $x^4-\frac{1}{x^4}=(x^2+\frac{1}{x^2})(x^2-\frac{1}{x^2})=7\ast 3\sqrt{5}=21\sqrt{5}$